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The key phrase there in Koheles is "under the sun." The Gemara (Shabbos 30b) explains that this means the natural world, as opposed to Torah which was created "before the sun." (Rashi to Eccles. 1:3 explains a little differently: תחת can also mean "instead," and "sun" symbolizes the light of Torah, so תחת השמש means "things done instead of Torah study.")
...

The context of that passage does not refer to Biblical corruption. The general chapter in Jeremiah concerns God's expressed frustration at the people for not listening to Jeremiah's prophesies. It is in this context that the scriptural passage appears, which is:
אֵיכָה תֹאמְרוּ חֲכָמִים אֲנַחְנוּ וְתוֹרַת יְהֹוָה אִתָּנוּ אָכֵן הִנֵּה לַשֶּׁקֶר עָשָׂה ...

The Gemara in Taanit 28b quotes the verse in Yirmiyahu and resolves the contradiction by saying that the walls of Jerusalem were breached on the 17th day of the month of Tammuz during the time of the second Temple not the first Temple.

The Yerushalmi (Taanis 4:5) offers a different view: the walls were indeed breached on the 17th of Tammuz, but there was "a mixup of calculations" and it was mistakenly thought to be the 9th. (The implication, then, is that Hashem told the prophet to record the date that people thought it was.)
It then also goes on to note that there is a verse (Ez. 26:1) ...

The verse separates clearly between Israel and Judah -- the standard language referring to the Northern and Southern (Davidic) kingdoms:
Israel had committed adultery, I had put her away and given her a bill of divorcement
that yet treacherous Judah her sister feared not; but she also went and played the harlot
So it seems Israel ends up with a divorce ...

God takes the children of Israel back. No other nation replaces them:
http://ohr.edu/tw/5756/devarim/haazinu.tw
In times to come, when Israel is redeemed from among the nations and
Hashem gathers us to Him, Israel will say "Master of the Universe,
it's written in Your Torah that when a man divorces his wife and
banishes her from his life, should ...

The following is based on Rabbi Aryeh Kaplan's understanding of the history as presented in his introduction to: The Torah Anthology - Book of Esther (translation of Yalkut Me'am Loez by Rabbi Aryeh Kaplan). It is based in part on Megila 11b-12a.
Versions of Yirmiyahu's prophecy: (said in 3331 - 460 BCE)
וְהָיְתָה כָּל הָאָרֶץ הַזֹּאת לְחָרְבָּה ...

The Haftorah is very moving and probably hold the record for the most Jewish songs from one Haftorah! It especially relates to the Rosh Hashanah theme of Zichronos.
The Haftorah discusses God bringing the redemption, which may connect to the theme of Zichronos, as it involves God 'remembering' the Jews. Next is the scene of Rachel weeping and God promising ...

Yirmiyahu - Jeremiah - Chapter 8
How do you say, "We are wise, and the Law of the Lord is with us"? Verily, behold it is in vain, he made a false scribes' pen.
ח. אֵיכָה תֹאמְרוּ חֲכָמִים אֲנַחְנוּ וְתוֹרַת יְהֹוָה אִתָּנוּ אָכֵן הִנֵּה לַשֶּׁקֶר עָשָׂה עֵט שֶׁקֶר סֹפְרִים:
(Rashi translation)
Verily, behold it is in vain: Behold your wisdom is in ...

The Mahari Kara also says all three t'rumos are called "reshis". I think they're probably
t'ruma g'dola (called "reshis" in Bamidbar 18:12 and D'varim 18:4),
t'rumas maaser (never explicitly called "reshis" that I know of, but it's called "t'ruma" in Bamidbar 18:26 and compared to t'ruma g'dola in Bamidbar 18:27), and
chala (called "reshis" and "t'ruma" in ...